Strain distribution through a propagating Lüders band front

Scripta METALLURGICA

Vol. S, pp. 213-216, 1971 Printed in the United States

Pergamon Press~ Inc.

STRAIN DISTRIBUTION THROUGH A PROPAGATING LUDERS BAND FRONT* D. W. Moon Department of Mechanical Engineering University of California, Davis

(Received January 15, 1971) Introduction The mechanism of propagation of a LUders band has been shown by several authors to be a hetergeneous process at the band front (1,2,3).

On a scale as small as the grain diameter or

less, the propagation is certainly a discontinuous process of dislocation a c t i v i t y .

However,

there is much information and understanding to be obtained from the macroscopic strain d i s t r i bution through the band front under both static and dynamic conditions.

A LUders band front

propagating in a uniform specimen at a constant velocity, as represented by Figure l , has been treated by Hart (4).

He has shown that the true strain rate (~) can be expressed as : VB ( ~ )

,

(1)

where VB is the propagation velocity of the front, c is the true strain, and ~ is measured in a

reference system moving with the band front. CLIP GAUGE

At

'~v B

TB

P

SPECIMEN e

i I~O

X=L

x

STRAIN ON T~'ECIMEN AXIS

FIG. l Schematic representation of a specimen and associated strain distribution along the axis for a single propagating LUders band front.

This research was supported in part by the National Science Foundation.

213

I,

214

PROPAGATING

LUDERS BAND FRONT

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3

On an average value basis, equation {l) must be true; however,~_only VB can be readily measured in thelaboratory.

The slope of the propagating front (~-~) is of major interest and has been

very elusive to experimental observations (5). The purpose of this paper is to present a method for directly measuring (~-~) on a dynamic basis. o

Clip Gauge Measurement Consider a single front propagating in a uniform cross-section specimen at a constant velocity (VB). The output of a clip gauge ahead of the front will be the elastic strain plus any pre-yield microstrain for the specimen (6). As the front moves into the gauge length L, the output of the clip gauge will be the integrated average strain on L and can be schematically represented by the curve in Figure 2. Zone I

Zone II

I tI

Zorn III

! t2

I I I

Zone IV

I I

Zor~ V

t4 TlUE

FIG. 2 Schematic representation of a clip gauge output as a single front sweeps by the gauge. The time t I is that at which the front f i r s t reaches point A on Figure l ; t 2 is the time at which the front has passed A; t 3 represents the arrival of the front at B; t 4 is the time at which the front has passed B. From t 2 to t 3 the moving front is completely within the gage length and adding strain at a constant rate VB~L/L, where cL is the plastic LUders strain. The output of the clip gauge is proportional to the total change of length AL within the gage length

L.

Therefore, the total strain is e = (~)+ microstrain + ~(~) + CLVB At/L

(2)

in which the f i r s t term is the elastic strain, E(~) is the strain associated with the band front, and ELVBAt/L is the LUders strain behind the band front within the gauge length. The difference between true strain (c) and engineering strain (e) has been ignored since the LUders strain is usually only a few percent. The output of the clip gauge will be the sum of these terms and can be divided into "time zones" as shown in Figure 2. The elastic strain and microstrain can be assumed constant and will drop out of consideration in the following development. The time derivative of the gauge output is

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S, No. 3

PROPAGATING

d ~

LUDERS BAND FRONT

EL VB (e) : [ ~ c(~) + -~C-~] ,

215

(3)

and must be considered separately in each of the time zones shown in Figure 2. In Zone I (4)

d (e) = 0; t < t l BT In Zone I I d

d-t (e) =

[E(C)]; t I < t < t 2

(5)

In Zone I I I ~L VB

d (e) = T ; BT

t2 < t < t3

(6)

In Zone IV EL VB d (e) = d d-t ---[-- " ~

[E(~)]; t 3 < t < t 4

(7)

In Zone V d

(e) : 0; t > t 4.

(8)

The most unique feature of the propagating LUders band treated here is found in Zones I I and IV.

The l e f t side of each equation (4)-(8) is simply the time derivative of the c l i p gauge

output and can be obtained graphically from s t r i p charts after a test, or may be obtained during a test by electronically d i f f e r e n t i a t i n g the c l i p gauge output.

The r i g h t side of equation

(5) contains the slope of the LUders band front as given by Hart and shown in equation (1). The r i g h t side can be written as d [E(~)] : r d e l l d ~ d-t 'd~' 'dt" " But since

dZ ~ = x - VBt, then ~ =

(9)

- VB and with rearrangement, dE

~:

l

- ~

d

(e) .

(IO)

I f we define

- l~ d then, holding

(e) -= G ( t , ~)

(ll)

~ constant and integrating on time t2

I G (t,

)Idt :EL

tI

L=const.

Equation (12) is the normalizing equation which determines the appropriate scale factors or the gain of the electronic c i r c u i t s and allows the amplifiers to be suitably adjusted.

(12)

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LUDERS BAND FRONT

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Similarly, the strain distribution of the propagating front is given by w

I G(t, IId = o

in which t front.

(131

t=const.

is held constant and the integration is carried out over the width

w of the

While i t is clear that this treatment does not yield an analytic expression for c(~),

the propagating LUders band front can be measured and plotted under conditions of dynamic propagation. Bonded Strain Gauge Measurement The propagation of the LUders band front under a bonded type strain gauge will not, in general, cause the same output as the clip gauge. The clip gauge is essentially a point contact device whereas the bonded gauge is a surface integrating device. The output of the bonded gauge circuitry will be dependent on the angle between the tensile axis and the surface trace of the propagating front. I f this angle is 90° , the results will be the same as those shown for the clip gauge above. For an angle less than 90° , Figure 2 will be modified by the geometric relation between the propagating front and the active portion of the strain gauge. The same strain zones will be clearly defined as long as the strain gauge dimensions are large with respect to the band front width. The use of a bonded strain gauge provides the same results in Zones I, I l l , and V as the use of the clip gauge. However, there is significant departure from clip gauge results in Zones II and IV. This departure will be due to the interaction of the propagating band front with the leading edge of the bonded gauge. Thus, the useful information to be obtained is obscured. The bonded gauge technique does not lead to precise information on the propagating strain profile. Conclusions This paper has shown that the strain profile for a slowly (well below acoustic velocities) propagating LUders band can be directly measured. As shown by equations (4), (6), and (8), the usual assumption that all strain is confined to the propagating front can be readily verified by simple experimental techniques. The strain gradient in the LUders band front is shown to be directly related to the time derivative of the clip gauge output in Zones II and IV. From Zone I I , a simple method is shown for obtaining the shape of the band front strain distribution. References I.

D. W. Moon and T. Vreeland, Jr., Acta Met. ]]7, 989 (1969).

2.

G. T. Hahn, Acta Met. lO, 727 (1962).

3.

W. Sylwestrowicz and E. O. Hall, Proc. Phys. Soc. Lond. B64, 495 (1951).

4.

E. W. Hart, Acta Met. 3, 146 (1955).

5.

J. F. Butler, J. Mech. Phys. Solids lO, 313 (1962).

6.

T. Vreeland, Jr., D. S. Wood, and D. S. Clark, Acta Met. ~, 414 (1953).